L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.120 − 1.72i)3-s + (−0.499 − 0.866i)4-s + (0.181 + 0.314i)5-s + (1.55 + 0.759i)6-s + (2.51 − 4.36i)7-s + 0.999·8-s + (−2.97 + 0.416i)9-s − 0.363·10-s + (1.40 − 2.42i)11-s + (−1.43 + 0.968i)12-s + (1.98 + 3.44i)13-s + (2.51 + 4.36i)14-s + (0.521 − 0.351i)15-s + (−0.5 + 0.866i)16-s − 6.69·17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.0696 − 0.997i)3-s + (−0.249 − 0.433i)4-s + (0.0812 + 0.140i)5-s + (0.635 + 0.310i)6-s + (0.952 − 1.64i)7-s + 0.353·8-s + (−0.990 + 0.138i)9-s − 0.114·10-s + (0.422 − 0.731i)11-s + (−0.414 + 0.279i)12-s + (0.550 + 0.954i)13-s + (0.673 + 1.16i)14-s + (0.134 − 0.0908i)15-s + (−0.125 + 0.216i)16-s − 1.62·17-s + ⋯ |
Λ(s)=(=(666s/2ΓC(s)L(s)(−0.0351+0.999i)Λ(2−s)
Λ(s)=(=(666s/2ΓC(s+1/2)L(s)(−0.0351+0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
666
= 2⋅32⋅37
|
Sign: |
−0.0351+0.999i
|
Analytic conductor: |
5.31803 |
Root analytic conductor: |
2.30608 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ666(223,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 666, ( :1/2), −0.0351+0.999i)
|
Particular Values
L(1) |
≈ |
0.808328−0.837296i |
L(21) |
≈ |
0.808328−0.837296i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.5−0.866i)T |
| 3 | 1+(0.120+1.72i)T |
| 37 | 1+T |
good | 5 | 1+(−0.181−0.314i)T+(−2.5+4.33i)T2 |
| 7 | 1+(−2.51+4.36i)T+(−3.5−6.06i)T2 |
| 11 | 1+(−1.40+2.42i)T+(−5.5−9.52i)T2 |
| 13 | 1+(−1.98−3.44i)T+(−6.5+11.2i)T2 |
| 17 | 1+6.69T+17T2 |
| 19 | 1−4.01T+19T2 |
| 23 | 1+(2.61+4.53i)T+(−11.5+19.9i)T2 |
| 29 | 1+(3.09−5.36i)T+(−14.5−25.1i)T2 |
| 31 | 1+(2.38+4.12i)T+(−15.5+26.8i)T2 |
| 41 | 1+(−0.969−1.67i)T+(−20.5+35.5i)T2 |
| 43 | 1+(0.533−0.924i)T+(−21.5−37.2i)T2 |
| 47 | 1+(−2.40+4.16i)T+(−23.5−40.7i)T2 |
| 53 | 1−8.18T+53T2 |
| 59 | 1+(3.74+6.48i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−0.597+1.03i)T+(−30.5−52.8i)T2 |
| 67 | 1+(2.35+4.08i)T+(−33.5+58.0i)T2 |
| 71 | 1−0.867T+71T2 |
| 73 | 1+7.28T+73T2 |
| 79 | 1+(−4.29+7.44i)T+(−39.5−68.4i)T2 |
| 83 | 1+(5.12−8.87i)T+(−41.5−71.8i)T2 |
| 89 | 1−16.1T+89T2 |
| 97 | 1+(7.87−13.6i)T+(−48.5−84.0i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.50906798213104942628100712883, −9.066528532878749892544227588851, −8.424070223600582542800241331233, −7.53549398852154690967042809015, −6.82944086904077452892662157402, −6.24820674677084733398294324246, −4.85722982604022905847331053230, −3.85338460690090373462779535440, −1.91720193321941790790997986872, −0.73329209591714173451359218478,
1.84706988655242206325499761994, 2.96732708966190618784736774459, 4.22539131992868549465825980338, 5.20744891745346969144047572186, 5.86323890536383235690623560178, 7.53752662430550010253085925531, 8.659546890945247594735483343210, 9.000177669269998759480488223914, 9.782985827477241361526784223506, 10.83247308134525509193031634237